Electromagnetic-Power-based Characteristic Mode Theory for Perfect Electric Conductors

Transcription

1 R. Z. LIAN: EM-BASED CMT FOR EC Electromagnetic-ower-based Characteristic Mode Theory for erfect Electric Conductors Renzun Lian Abstract In this paper, a new way to normalize various electromagnetic powers of EC systems and a new way to decompose power quadratic matrix are introduced, and then an ElectroMagnetic-ower-based Characteristic Mode Theory (CMT for EC systems (EC-EM-CMT is built. The EC-EM-CMT is not only valid for the EC systems which are placed in vacuum, but also suitable for the EC systems which are surrounded by any electromagnetic environment. Based on the new normalization way and the new decomposition way, the complex characteristic currents and non-iative Characteristic Modes (CMs can be constructed; some titional concepts, such as the system ut impedance and modal ut impedance etc., are redefined based on the language of field instead of the language of circuit; the titional characteristic quantity, Modal Significance (MS, is generalized; a series of new power-based CM sets, such as the Radiated power CM (RaCM set and Output power CM (OutCM set etc., are introduced. It is proven in this paper that various power-based CM sets of a certain objective EC structure are independent of the external environment and excitation; the OutCM set has the same physical essence as the CM set derived from the titional Surface EFIE-based CMT for EC systems (EC-SEFIE-CMT, but the former can be viewed as the generalization of the latter; the non-iative space constituted by all non-iative modes is identical to the interior resonance space constituted by all interior resonant modes of closed EC structures, and the non-iative CMs constitute a basis of the space. Based on above these, the normal Eigen-Mode Theory (EMT for closed EC structures and the EC-SEFIE-CMT are classified into the EC-EM-CMT framework, and then the ideas of EC-SEFIE-CMT are liberated from its titional framework, MoM. In addition, a variational formulation for the external ttering problem of EC structures is provided in this paper, based on the conservation law of energy. Index Terms Characteristic Mode (CM, Characteristic Quantity, Input Impedance, Input ower, Modal Expansion, Modal Impedance, Output ower. T I. INTRODUCTION E Theory of Characteristic Mode (TCM, or equivalently called as Characteristic Mode Theory (CMT, was firstly introduced by Robert J. Garbacz [], and subsequently refined by Roger F. arrington and Joseph R. Mautz under the Integral Equation-based MoM (IE-MoM framework. In 97, arrington and Mautz built their CMT for EC systems based R. Z. Lian is with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 67, China. ( rzlian@vip.6.com. on the Surface EFIE-based MoM (EC-SEFIE-CMT []-[]. Afterwards, some variants for the EC-SEFIE-CMT were introduced under the IE-MoM framework, such as the Volume Integral Equation-based CMT for Material bodies (Mat-VIE-CMT [4], the Surface Integral Equation-based CMT for Material bodies (Mat-SIE-CMT [5], the Surface MFIE-based CMT for EC systems (EC-SMFIE-CMT [6], and the Surface CFIE-based CMT for EC systems (EC-SCFIE-CMT [7]. Recently, the EC-SEFIE-CMT is re-derived from complex oynting s theorem in [8], and the power characteristics of the Characteristic Modes (CMs constructed by Mat-SIE-CMT is researched in [9], such that the physical pictures of the EC-SEFIE-CMT and Mat-SIE-CMT become clearer. In fact, the CMT is a new modal theory which is essentially different from the titional Eigen-Mode Theory (EMT []-[] in mathematical physics. The EMT is a source-free modal theory or called as passive modal theory, but the CMT is an active modal theory. In mathematical physics, the EMT is divided into two categories, the normal EMT (such as the passive modal problems of closed structures [] and the singular EMT (such as the passive modal problems of open structures []. Generally speaking, the active modal theory is more suitable for analyzing the open electromagnetic systems, because the Sommerfeld s iation condition [] at infinity signifies a continuous energy put from the open system, and then a non-zero external excitation is necessary for the system to sustain a stationary oscillation of field. Because of these above, the CM-based analysis and design methodologies for various open electromagnetic systems, especially for iation systems, have got more and more attentions in electromagnetic engineering society, and a comprehensive review for the CMT and its applications in antenna community can be found in [8]. In this paper, a new way to normalize various electromagnetic powers of EC systems and a new way to decompose power quadratic matrix are introduced, and then an ElectroMagnetic-ower-based CMT for EC systems (EC-EM-CMT is established. The EC-EM-CMT is not only valid for the EC systems which are placed in vacuum, but also suitable for the ones which are surrounded by any electromagnetic environment. Based on the new normalization way and the new decomposition way, the complex characteristic currents and non-iative CMs can be constructed; some titional concepts, such as the system ut impedance and modal ut impedance etc., are redefined based on the language of field instead of the language of circuit; the titional characteristic

2 R. Z. LIAN: EM-BASED CMT FOR EC quantity, Modal Significance (MS [8], is generalized; a series of power-based CM sets, such as the Radiated power CM (RaCM set and Output power CM (OutCM set etc., are introduced. The various power-based CM sets of a certain objective EC structure are independent of the external environment and excitation. Among these CM sets, the OutCM set has the same physical essence as the CM set derived from EC-SEFIE-CMT, but the former can be viewed as the generalization of the latter, and then the ideas of EC-SEFIE-CMT are liberated from its titional framework, MoM. It is proven in this paper that all non-iative modes constitute a linear space called as non-iative space, and the space is identical to the interior resonance space constituted by all interior resonant modes of closed EC structures, and the non-iative CMs derived in EC-EM-CMT constitute a basis of the space. Based on above these, the normal EMT for closed EC structures [] is classified into the EC-EM-CMT framework. In addition, a variational formulation for the external ttering problem of EC structures is provided in this paper, based on the conservation law of energy [4]. This paper is organized as follows. Sections II-VII give the priples and formulations related to EC-EM-CMT, and some necessary discussions are provided in Sec. VIII. Section j t IX concludes this paper. In what follows, the e ω convention is used through. II. VARIOUS ELECTROMAGNETIC OWERS AND TEIR NORMALIZATIONS In this paper, the power characteristics of a EC system which is placed in an arbitrary time-harmonic electromagnetic environment is studied. When an external excitation is impressed, there exist three kinds of fields in whole space, that are the F im generated by impressed excitation, the F en sys generated by external environment, and the F generated by EC system, as illustrated in Fig., here F E,. The term time-harmonic electromagnetic environment means that the en F is a time-harmonic electromagnetic field operating at the im same frequency as F. Based on the superposition priple [4], the F sys is considered as the ttering field due to the ident field im en F F F, because the EC system is regarded as a whole object in this paper. The F sys is equivalently written as F F J, and the EC system is simply called as tterer. The J is the surface ttering current on the tterer, and the symbol F J represents the field generated by J in vacuum, and the mathematical expression for this operator can be found in []. A. Various electromagnetic powers. The power done by F on J is just the ut power for tterer from both impressed excitation and external environment, and its mathematical expression is as follows F ( J J J ( J J j Wm J We J ω In (, the coefficient is due to time average; the ω π f is angle frequency, and the f is frequency; the inner product is defined as f, g f g dω Ω, and the symbol is Ω the dot multiplication of field vectors, and the superscript represents complex conjugate; the V is the whole boundary of tterer V. The second and third equalities in ( respectively originate from the SEFIE [] and complex oynting s theorem [5]. In (, the, W m, and W e are respectively the iated power carried by F and the al magnetically and electrically red energies of F, and [5] ( J E ( ds E η η, Wm ( J, μ 4 We ( J ε E 4 ( (. (. (. here the relations ( η nˆ E and E η ˆ n on S have been employed in the (., and the η μ ε and ˆn are respectively the wave impedance in vacuum and the er normal of S ; the S is a closed spherical surface at infinity; the ε and μ are respectively the permittivity and permeability in vacuum. In fact, the system put power is as follows ω m e J ( J J J j W J W J sys J F EC System (Objective Scatterer en F im F Fig.. Various fields generated by various sources. and the physical essence of the second equality in ( is just the conservation law of energy [4], i.e., Environment Impressed Source ( J J (4 The physical meanings of the real and imaginary parts of are respectively the active and ive powers due to the

3 R. Z. LIAN: EM-BASED CMT FOR EC interaction between F and tterer. The physical meanings of the real and imaginary parts of are respectively the iated power and ively red power generated by J. To simplify the symbolic system and to consider of the physical meanings of various powers mentioned above, the Re{ } and Im{ } are simply denoted as act and respectively, and the Re{ } and Im{ } are simply denoted as and respectively, and the is simply called as red power, so { } Re{ } { } { } ω Re act (5. Im Im W W (5. m e Obviously, there exist above equivalent relations between the symbolic systems {,, act ; ut, active, ive } and {,, ; put, iated, red }, and the two symbolic systems have their own merits to depict the characteristics of tterer from different aspects. The former symbolic system focuses on emphasizing the interaction between the F and tterer, whereas the latter symbolic system focuses on emphasizing the aspects of tterer s ability to put various electromagnetic energies. In the following discussions, the latter symbolic system is utilized. B. The normalizations for various electromagnetic powers. The value of can be uniquely obtained from the J, and this is why the is expressed as the one-variable function of J in (. Because the J ( r is a complex vector function of the position r, here r V, so the depends on both the amplitude of J and the distribution of J. The distribution of J ludes three aspects, that are the distribution of J, the distribution of the phase of J, and the distribution of the polarization of J [8]. To eliminate the influence from the amplitude and to take account of the distribution and to consider of that (, the is normalized as follows their dimensions are Ohm, so they can also be called as system put impedance, iated resistance, and red ance respectively, and then they can be equivalently denoted as Z, R, and X as follows ( ( ( ( Z J J (7. R J J (7. X J J (7. In addition, considering of the (4 and (5, the Z, R, and X can also be called as system ut impedance, resistance, and ance respectively. III. TE MATRIX FORMS FOR VARIOUS OWERS The J can be expanded in terms of the basis function set as follows { b } J ( a ab Ba (8 T here B b, b,, b and a [ a, a,, a ], and the superscript T represents the transpose of matrix. The symbol in (8 represents ordinary matrix multiplication. Inserting (8 into the second equality in (, the power J can be rewritten as the following matrix form. a a a (9 The elements of matrix are pζ ( b ( bζ for any ζ,,,,, and the matrix can be decomposed as follows here ( J ( J ( J ( J, J J j J ( J,, ( J ( J,, ( J (6. ( J, J (6. ( J ( J ( J,, ( J ( J, J (6. ( J here j (. j (. The superscript represents the transpose conjugate of matrix. It is obvious that the matrices and are ermitian, so the numbers a a and a a are real for any a [6], and then (. (. a a a a a a The,, and in (6 are respectively called as the normalized system put, iated, and red powers. In fact, and

4 R. Z. LIAN: EM-BASED CMT FOR EC 4 Z a R a X a ( a a a a J a, a, ( a a a a J a, a, ( a a a a J a, a, (. (. (. here a J a ( J ( a, J ( a, and the elements of J are jζ ( b, bζ. Obviously, the matrix is either positive definite or positive semi-definite, and the matrix J is ermitian and positive definite. For a certain linear EC system, all currents which can be excited on it constitute a linear space called as current space; correspondingly, the expansion vectors constitute a vector space; for a certain { b }, the current space and the vector space are isomorphic, so they are not specifically distinguished from each other in the following sections. The space has many different basis sets, and every basis set can span whole space. The main destination of EC-EM-CMT is to select an electromagnetic power as the objective power, and then to search for a special basis set, such that the objective power corresponding to this special basis set has some desired characteristics. The elements in this special basis set are called as modal currents/vectors or characteristic currents/vectors, and the corresponding fields are called as modal fields or characteristic fields. The modal currents, vectors, and fields are collectively referred to as CMs. In fact, the EC-EM-CMT is an object-oriented theory, because the selection for the objective power is relatively flexible. For example, the normalized system iated power R( J, the normalized system red power X ( J, and the system put power ( J are respectively selected as the objective powers in the following Secs. IV, V, and VI. owever, it must be clearly pointed that the CM sets constructed for different objective powers are not necessarily identical to each other, so the different superscripts are added to different sets to distinguish them from each other. IV. RADIATED OWER CARACTERISTIC MODE (RACM SET AND RACM-BASED MODAL EXANSION In fact, the normalized iated power R( a quantitively reflects the iation ability of the distribution of J ( a. The CM set which spans whole space and can provide the extremes of R( a is constructed in this section. A. Radiated power CM (RaCM set. The matrices and J are ermitian, and the J is positive definite, so the necessary condition to achieve the extremes of R( a is following generalized characteristic equation [6]. a λ J a ( here,,,, and the characteristic values are real [6]. The modal currents and fields are as follows, ( (4., (4. J r B a r V F r F J r for any,,,. The iated modal powers and their normalized versions are as follows ( ( a, a a λ J a a a (5. R (5. for any,,,. As proven in the Appendix of this paper, the coupling iated modal powers satisfy the following orthogonality for any ζ,,,,., δζ ( a aζ E ( ds S ζ E ζ (6 η η, ζ here the δ ζ is Kronecker delta symbol. The CMs derived above are specifically called as Radiated power CMs (RaCMs to be distinguished from the other kinds of CMs constructed in Secs. V and VI. Because, for any,,,, all RaCMs can be divided into two categories, that are the iative RaCMs corresponding to, > ( r, r,, r R and the non-iative RaCMs corresponding to, ( n, n,, nn, here { r, r,, rr} { n, n,, n N}, and { r, r,, rr} { n, n,, n N} {,,, }. B. RaCM-based modal expansion. Because of the completeness of basis function set { b } and the characteristic vector set { a } [6], the ttering vector a, the ttering current J, and the ttering field F can be expanded in terms of the RaCM set as follows a c a (7., ( (7. J r c J r r V, ( F r c F r r (7. Based on the orthogonality in (6, the system iated power can be expanded in terms of the iated modal powers as, a a c (8

5 R. Z. LIAN: EM-BASED CMT FOR EC 5 C. Expansion coefficients. Based on the discussions in Sec. II, the written as follows { } J can be J Re J Re J (9 Because of the ( and (9, the J B a on tterer will make the following functional be zero and stationary [7]. F ( a Re Ba ( a a ( Inserting the (7. into ( and employing the Ritz s procedure [8], the following simultaneous equations for the expansion coefficients { c } are derived for any,,,. Re Ba ζ ζ ζ ζ ζ ζ a c a c a a Im Ba ζ ζ ζ ζ ζ ζ j a c a j c a a (. (. By employing the orthogonality in (6, the ( can be rewritten as the following ( for any,,,. Re J c a a c a a Im J j c a a j c a a i.e., c ( c, Re J c ( c, j Im J (. (. (. (. Because the, are real numbers, the two equations in ( can be uniquely written as following one for any,,,., c J (4 when the matrix is positive definite When the matrix is positive definite,, > for any,,,, and then all c can be explicitly expressed as c J (5, The symbol in (5 is the ordinary lar multiplication. when the matrix is positive semi-definite When the matrix is positive semi-definite, the iative RaCMs corresponding to, > ( r, r,, r R and the non-iative RaCMs corresponding to, ( n, n,, nn simultaneously exist in the RaCM set. The expansion coefficients of the iative RaCMs can also be expressed as (5, but the coefficients of the non-iative RaCMs cannot. In fact, the non-iative RaCMs are just the interior resonant modes of closed EC structures as proven in Sec. VIII-A, and the interior resonant modes should not be luded in the expansion formulation for external ttering problems [], and then the expansion coefficients of all non-iative RaCMs are zeros for external ttering problems. V. STORED OWER CARACTERISTIC MODE (STOCM SET AND STOCM-BASED MODAL EXANSION Similarly to the R( a, the normalized red power X( a quantitively reflects the tterer s ability to ively re energies. The CM set which spans whole space and can provide the extremes of X( a is constructed in this section. A. Stored power CM (StoCM set. The matrices and J are ermitian, and the matrix J is positive definite, so the necessary condition to achieve the extremes of X( a is the following characteristic equation [6]. a J a λ (6 here,,,, and all characteristic values are real [6]. The modal currents and modal fields can be similarly obtained as (4. The ively red modal powers and their normalized versions are as follows for any,,,. ( ( a, a a λ J a a a (7. X (7. By doing some necessary orthogonalizations for the degenerate modes [6], the following orthogonality holds for any ζ,,,,., δζ ζ a a ω, μ ε E 4 4 ζ ζ (8 The proof for the orthogonality (8 is similar to the proof for (6 given in Appendix, and the proof for (8 will not be repeated in this paper.

6 R. Z. LIAN: EM-BASED CMT FOR EC 6 The modes derived above are specifically called as ively Stored power CMs (StoCMs to be distinguished from the other kinds of CMs constructed in Secs. IV and VI. All StoCMs can be divided into three categories, that are the inductive StoCMs corresponding to, > ( ι, ι,, Ι ι, the capacitive StoCMs corresponding to, < ( χ, χ,, χ Χ, and the resonant StoCMs corresponding to, ( ρ, ρ,, ρ Ρ. ere, { ι, ι,, Ι ι } { χ, χ,, χχ} { ρ, ρ,, ρρ } {,,, } ; the index set { ι, ι,, Ι ι }, the index set { χ, χ,, χ Χ }, and the index set { ρ, ρ,, ρ Ρ } are pairwise disjoint. In fact, the resonant StoCMs can be further divided into two categories based on their iated powers, that are the iative resonant StoCMs corresponding to the positive iated powers and the non-iative resonant StoCMs corresponding to the zero iated powers. As proven in Sec. VIII-A, the non-iative resonant StoCMs are just the interior resonant modes of closed EC structures, so they can also be called as interior resonant modes. Similarly, the iative resonant StoCMs can be also called as exterior resonant modes. B. StoCM-based modal expansion. Because of the completeness of basis function set { b } and the characteristic vector set { a } [6], the ttering vector a, the ttering current J, and the ttering field F can be expanded in terms of the StoCM set as follows a c a (9., ( (9. J r c J r r V, ( F r c F r r (9. Im Ba ζ ζ ζ ζ ζ ζ a c a c a a Re Ba j ( a c a j c a a ζ ζ ζ ζ ζ ζ (. (. By employing the orthogonality in (8, the ( can be rewritten as c ( c j, j Im J c ( c j, Re J (4. (4. Because the, is real, the two equations in (4 can be uniquely written as the following one for any,,,. j, c J (5 For any non-resonant StoCM,,, and then the expansion coefficients corresponding to the non-resonant StoCMs can be explicitly expressed as the following (6 for ι, ι,, ι, χ, χ,, χ. any Ι Χ c J (6 j, Based on the orthogonality given in (8, the system red power can be expanded in terms of the red modal powers as, a a c ( C. Expansion coefficients. Based on Sec. II, the J can be written as follows ( J Im J ( Based on the ( and (, the J B a on tterer will make the following functional be zero and stationary. F ( a Im, B a E a a ( Inserting the (9. into ( and employing the Ritz s procedure [8], the following simultaneous equations for the expansion coefficients { c } are derived for any,,,. For the iative resonant StoCMs (exterior resonant modes, there are not the concise expressions for their expansion coefficients. For the external ttering problems, the non-iative resonant StoCMs (interior resonant modes should not be luded in the expansion formulation [], so their expansion coefficients are zeros. VI. OUTUT OWER CARACTERISTIC MODE (OUTCM SET AND OUTCM-BASED MODAL EXANSION The CM set which satisfies the put power orthogonality in (45 has some very good properties, and it is constructed in this section, and it is specifically called as the Output power CM (OutCM set to be distinguished from the other kinds of CM sets constructed in the Secs. IV and V. It must be clearly pointed here that the OutCM set has the same physical essence as the CM set derived from EC-SEFIE-CMT []-[]. owever, the OutCM set is more advantageous in some aspects, as discussed in the Sec. VIII-D. A. Output power CM (OutCM set. The matrices and are the functions of frequency f, so they are denoted as ( f and ( f for the convenience of

7 R. Z. LIAN: EM-BASED CMT FOR EC 7 following discussions. The procedures to construct OutCM set will be provided by separately focusing on two different cases: case I, there doesn t exist any non-iative mode at frequency f, and then the ( f is positive definite; case II, there exist some non-iative modes at frequency f, and then the ( f is positive semi-definite. To construct the OutCM set, when the matrix ( f is positive definite. Both ( f and ( f are ermitian, and the ( f is positive definite, so there is a vector set { a }, such that [6] a ( f ( f aζ ( f, ( f δζ (7. a ( f ( f aζ ( f, ( f δζ (7. for any ζ,,,,, and then a ( f ( f aζ ( f ( f δ ζ (8 for any ζ,,,,, here (9 f f j f,, for any,,,. The characteristic vector set { a } can be obtained by solving the following (4 and doing some necessary orthogonalizations for the degenerate modes [6]. λ (4 f a f f f a f here the λ is real [6]. The modal currents and modal fields can be similarly expressed as the (4 for any,,,. The modal put power is given in (9, and its normalized version (modal put impedance is as follows here Z f R f j X f R X ( f ( f a f f a f a f J f a f a f f a f a f J f a f a f f a f a f J f a f (4. (4. (4. The, ( f and R ( f are respectively called as modal iated power and modal iated/put resistance; The, ( f and X ( f are called as modal red power and modal red/put ance respectively. In fact, it is obvious that the characteristic values λ ( f derived from (4 satisfy the following relation (4 for any,,,, ( f X ( f λ ( f (4 ( f R ( f, and the characteristic equation (4 is just the necessary condition to achieve the extreme of following functional [6]. F ( a a X a a a a R a a a (4 To construct the OutCM set, when the matrix ( f is positive semi-definite. When the matrix ( f is positive semi-definite, there doesn t exist a ready-made mathematical theorem to support that the vector set { a } satisfying (7-(8 can be obtained by solving (4. In this paper, a limitation method is employed to derive the { a } at f, and it ludes two core steps: step I, to find the f at which the ( f is positive semi-definite; step II, to construct the OutCM set at f. Step I, to find the f Due to the limitation viewpoint, if the ( f is positive semi-definite, some iative modes at f will approach to the non-iative modes at f, when f f. Because the non-iative mode is also the resonant mode as proven in Sec. VIII-A, its Z ( f equals to zero, and then the limitation of corresponding characteristic value λ ( f X ( f R ( f is an indeterminate form during f f [9], so it is sometimes difficult to efficiently recognize the f only based on the limitation lim f f λ ( f. In fact, the Z ( f, which equals to R ( f j X ( f, is a more complete description for the modal characteristic to utilize various energies than the λ ( f, because the λ ( f has only ability to depict the ratio of the X ( f to the R ( f, but the X ( f and R ( f themselves cannot be explicitly reflected by the λ ( f. Based on above observations, a modal impedance based method to search for the frequency f is developed here. The R ( f and X ( f are the one-variable real functions ab f, and their function curves can be easily obtained by using frequency sweep technique. The R ( f curve will pass through the point ( f,, if and only if the mode doesn t iate at the frequency f. Step II, to construct the OutCM set at f When the f is found, the modal vectors, currents, fields, powers, and impedances at f can be obtained as the following limitations for any,,,. ( f f (44. ( f f (44. ( f f (44. ( f f (44.4 (44.5 a f a f J f J f F f F f f f Z f lim Z f f f

8 R. Z. LIAN: EM-BASED CMT FOR EC 8 Summarizing both the positive definite ( f and positive semi-definite ( f cases. Based on the above discussions, the coupling modal put powers for any iative modes r, r,, r R and non-iative modes n, n,, nn satisfy the following put power orthogonality. δζ ( a aζ J ζ (45. and, δζ ( a aζ E ( ds S ζ E ζ η η, ζ δ a a, ζ ζ here ω, μ ε E 4 4 ζ ζ (45. (45.,, a a c (48. a a c (48. a a c (48. C. Expansion Coefficients. Based on the discussions in Sec. II, the J on tterer will make the following functional be zero and stationary. F4 J J E J E J,, (49 By inserting the (47. into (49 and employing the Ritz s procedure [8], the following simultaneous equations for the expansion coefficients { c } are derived for any,,,. J E J c E c J E,,, ζ ζ ζ ζ ζ ζ (5. J E J c E c J E,,, V ζ ζ ζ ζ ζ ζ (5. ( ( n n n j, r, r,, r,,,,,, R N (46 By employing the orthogonality in (45., the (5 becomes the following (5. In (46, the conclusion that the non-iative modes must be the resonant modes has been used. The proof for (45 is similar to the proof for (6 given in Appendix. That the symbols, and, are not respectively written as the, in Sec. IV and the, in Sec. V is to emphasize that the, and, are the real and imaginary parts of the modal put power, whereas the, and, are just the modal powers themselves derived by optimizing R( a and X ( a. In fact, this is just the reason why the, is called as modal iated power in this section, instead of being called as the iated modal power like the, in Sec. IV. Other terms can be similarly explained. B. Modal expansion. Similarly to the Secs. IV and V, the ttering vector a, the ttering current J, and the ttering field F can be expanded in terms of the OutCM set as follows a c a (47., ( (47. J r c J r r V, ( F r c F r r (47. Based on the power orthogonality (45, various system powers can be expanded as follows c J (5. c (5. ( for any,,,. when the matrix is positive definite When the matrix is positive definite, ( a a > for any,,,, and then for any,,,. Based on this, by solving (5 the expansion coefficients can be expressed as following (5 for any,,,. { c } c J (5. c (5. The symbol in (5. is the ordinary lar multiplication. It is obvious that the (5. is the non-physical solution for the non-zero external excitation E, so only the (5. is the desired physical solution. when the matrix is positive semi-definite When the matrix is positive semi-definite, the iative OutCMs corresponding to ( r, r,, r R and the non-iative OutCMs corresponding to ( n, n,, nn exist simultaneously. The iative OutCMs satisfy (5., and the non-iative OutCMs satisfy (5.. owever, only based on the (5., the expansion coefficients for the non-iative

9 R. Z. LIAN: EM-BASED CMT FOR EC 9 OutCMs cannot be determined, because all non-iative OutCMs (that are the interior resonant modes belong to the zero space of EFIE operator []. In fact, for the external ttering problem of the EC structures whose boundaries are closed surfaces, the interior resonant modes should not be luded in the modal expansion formulation, so the expansion coefficients of all non-iative OutCMs are zeros for the external ttering problem. VII. TE CARACTERISTIC AND NON-CARACTERISTIC QUANTITIES OF OUTCMS In Sec. VI-B, the system electromagnetic quantities (such as J, F, and are expanded in terms of series of modal components (such as c J, c F, and c, and to consider of the weight of every modal component in whole expansion formulation is important. owever, the components are either the complex vectors (such as c J and c F or the complex lars (such as c, so the modal weight cannot be directly derived from the modal components themselves, and then to establish an appropriate mapping from the modal index set { } to a real number set is necessary. In this section, only the OutCM-based modal expansion for the external ttering problem is considered. A. An illuminating example. Let us consider the following example at first. xˆ yˆ xˆ yˆ xˆ yˆ (5 Obviously, using the following mapping (54. to depict the weights of term ( xˆ yˆ and term ( xˆ yˆ is unreasonable. { ( xˆ yˆ, ( xˆ yˆ } {, } (54. quantitively depict the modal weights in whole modal expansion formulation. In this paper, the modal currents are normalized as follows J ( r, ( (56 ( J, J J r r V and then the modal fields and modal put powers are automatically normalized as follows ( r F F ( r, ( r (57 J, J (58 ( J, J and then the expansion coefficient (5. automatically becomes the following version for any iative OutCM. c J (59 Inserting the (58 and (59 into the OutCM-based modal expansion formulation for the in (48., the following equivalent version of (48. is obtained. J c (6 Obviously, the magnitude of term is unit just like the terms ( xˆ yˆ and ( xˆ yˆ in (55, and then two mappings can be established as the following (6 just like the (54.. here the symbol { ab, } represents the ordered array of numbers a and b. In fact, a reasonable mapping is as follows { ( xˆ yˆ, ( xˆ yˆ } { }, (54. because the (5 can be equivalently rewritten as the following (55. ( xˆ yˆ ( xˆ yˆ xˆ yˆ (55 here, { } { SMS sys } (6., { } { GMS sys } (6., SMS J GMS sys (6. (6. By comparing the (5 with (55, it is easy to find that the essential reason to lead to the inappropriate mapping (54. is that the terms xˆ yˆ and xˆ yˆ in (5 are not normalized. B. Modal normalization. Based on the example in Sec. VII-A, to normalize the OutCMs is necessary for establishing the appropriate mapping from the { } to a real number set whose elements C. The generalized Modal Significance (MS for system al power. Based on the (6-(6, it is easily found that: ( When a specific field F idents on tterer, only the first several terms, whose SMS are relatively large, are necessary to be luded in the truncated modal expansion formulation.

10 R. Z. LIAN: EM-BASED CMT FOR EC ( Generally speaking, when the sufficient terms, which have relatively large GMS, are luded in the modal expansion formulation, the truncated expansion formulation can basically coide with the full-wave solution for any external excitation. owever, it must be clearly pointed that this conclusion is not always right. For example, when the expansion formulation only ludes N terms, and the external excitation satisfies that F F N Mon V (here M >, the truncated expansion formulation cannot coide with the full-wave solution, because only the (NM-th modal component is excited, whereas this component is not luded in the truncated expansion formulation. Based on the discussions in above ( and (, it is obvious that the SMS and GMS quantitively depict the modal weight in whole modal expansion formulation, so the SMS is called as the Modal Significance (MS corresponding to the specific excitation or simply called as Special MS (SMS, and the GMS is called as the MS corresponding to general excitation or simply called as General MS (GMS. The SMS and GMS are collectively referred to as the generalized MS for system al power, and this is the reason why the superscripts are used in them. D. The generalized Modal Significance (MS for system active and ive powers. The real and imaginary parts of power is the active power act and the ive power. Because the in (6 is real, the following expansions can be derived. c { } act Re Re J c { } { } Im Im J c { } (6. (6. Similarly to the generalized MS for system al power, the following generalized MS for system active and ive powers are introduced to quantitively depict the modal weights in whole system active and ive powers. { } Re SMS SMS, Re act J E { } Re { } act GMS GMS Re for the active power, and { } Im SMS SMS, Im { } J E { } Im { } GMS GMS Im { } (64. (64. (65. (65. for the ive power. act / The reason to use the superscript sys in the SMS act / and GMS is the same as the reason to use the superscript sys in SMS and GMS. For the EC systems, the iated power is just the active part of ut power as illustrated in (5, however this conclusion doesn t hold for the material tterer case [5]. To maintain the consistency with the ElectroMagentic-ower-based CMT for act / Material bodies (Mat-EM-CMT [], the SMS and act / GMS are not respectively called as the SMS for system iated/red power and the GMS for system iated/red power, and this is just the reason to use the superscripts act instead of using the. E. The modal characteristic to allocate active and ive powers and the modal ability to couple energy from external excitation. From the above discussions, it is obvious that the modal ability to transform the energy provided by external excitation to a part of the system active and ive powers depends on the following three aspects: ( The modal ability to contribute system al power is quantitively depicted by the GMS defined in (6.. ( The modal characteristic to allocate the al modal put power to its active and ive parts is quantitively depicted by the Modal characteristic to Allocate modal Output ower (MAO defined as the following (66. MAO MAO mod, act mod, { } Re Im { } (66. (66. ( The modal ability to couple energy from external excitation is quantitively depicted by the Modal Ability to Couple Excitation (MACE defined as the following (67. mod MACE J (67 The superscripts mod in (66 and (67 are to emphasize that to introduce the MAO and MACE is to depict the characteristic of the mode itself, but not to depict the modal weight in whole system power. F. The characteristic and non-characteristic quantities. act mod, act Obviously, the GMS, GMS, and MAO are independent of the specific external excitation, so they are collectively referred to as modal characteristic quantities (or simply as characteristic quantities. owever, the SMS, act SMS, and MACE mod depend on the specific external excitation, so they are collectively referred to as modal non-characteristic quantities (or simply as non-characteristic quantities. The characteristic quantities and non-characteristic quantities are collectively referred to as modal quantities.

11 R. Z. LIAN: EM-BASED CMT FOR EC For the EC systems, the, and respectively, correspond to the modal far and near fields. Based on this, the following conclusions can be obtained. ( The generalized MS for system active and ive powers defined in (64 and (65 quantitively depict the weights of the modal far and near fields in whole system far and near fields. ( The MAO in (66 quantitively depicts the modal characteristic to allocate the modal field to its far and near field zones., ( To compare the MACE with the GMS sys, ( GMS sys act,, and GMS sys is an efficient method to judge if the specific excitation is suitable for exciting the objective EC system (the far field modes, and the near field modes. VIII. DISCUSSIONS In this section, some necessary discussions related to the theory developed in this paper are provided. A. The discussions for non-iative CMs. The following discussions in this subsection are suitable for the any kind of CM set developed in Secs. IV, V, and VI. When the matrix is positive semi-definite, a a if and only if a [6], and then a a and a a imply that ( αa βa ( αa βa for any complex numbers α and β. Based on this, it can be concluded that all a corresponding to ( a a a constitute a linear space [6], and this space is called as non-iative space in this paper. Any non-iative mode has a zero iation pattern at infinity due to the second equality in (., so its fields are zeros at any point of tterer based on the conclusions given in [], and then its tangential electric field is zero on whole V. This implies that Z a for any non-iative mode, here the Z is the impedance matrix derived from the Galerkin-based EFIE-MoM. The interior resonance space constructed by all interior resonant modes is identical to ( Z [], here the ( Z is the zero space of Z, and then any non-iative mode must be the interior resonant mode. In addition, it is obvious that any interior resonant mode is non-iative. Based on above these observations and considering of that ( Z, it can be concluded that the non-iative space, the interior resonance space, the zero space (, the zero space (, and the zero space ( Z are identical to each other, and all the CMs corresponding to ( a constitute a orthogonal basis of these spaces. Based on the above these, the normal EMT for closed EC structures is classified into the EC-EM-CMT framework. B. The discussions for various electromagnetic powers and various CMs. The al red energy { We, W m } ludes two parts, the,, non-iative/ive red energy { We, W m } and the,,,, iative red energy { We, W m }, and We Wm [5], so ω W W,, ( Wm We m e ω (68 This is just the common reason why the is called as ively red power (simply as red power in this paper, and why the subscript is utilized in the symbol. For the iation systems, the most desired operating state is,, that Wm We [5], [], so the ( J corresponding to non-zero J is usually desired in antenna engineering community. owever, that ( J is only a,, necessary condition to guarantee that Wm We instead of the sufficient condition, and an extreme example is the interior resonant state of EC cavity. In fact, this is just the reason why sometimes the strongest iation state and the resonant state cannot be simultaneously archived as explained in following Sec. VIII-C. Based on above observations, it is found that to minimize the absolute value of the functional F ( J in (4 is neither the necessary condition nor the sufficient condition to guarantee that the system operates at the strongest iation state, so it is valuable to develop the RaCM set for optimizing system iated power. For the wireless energy transfer systems [4], the most, suitable one in various energies is the W m, and the reasons, are that: ( the W m is not iative, so this kind of energy is red in near field zone and not dissipated, and then it is valuable for wireless energy transfer systems to rease, transmission efficiency; ( the W m is magnetic instead of being electric, so this kind of energy is more safe for creatures. In fact, the system s strongest state to re ive magnetic energy can be found in the StoCM set as discussed in Sec. V. Some aspects of the small antenna applications of the OutCM set, which has the same physical essence as the CM set constructed in EC-SEFIE-CMT []-[], have been comprehensively reviewed in [8], so they are not repeated here. In addition, the OutCM set constructed in this paper can also be applied to the electrically large antennas (such as the travelling wave antennas [5] and the non-iative EC structures (such as the EC cavities []. C. The relation between resonance and the strongest iation. Based on the formulations in (5, to maximize and to maximize act are equivalent to each other, i.e., the most strongest iation state of ttering current J and the most efficient way that the power is done by ident field E on tterer are equivalent to each other, and the latter state as a representative is discussed in this subsection. To efficiently distinguish the time domain quantities and the frequency domain quantities from each other, the time variable t is specifically added to the time domain quantities in this subsection. For any point r V, the time average of ( r, t J rt, E rt, is as follows

12 R. Z. LIAN: EM-BASED CMT FOR EC T ( r, t dt Re J i r Ei r T,, i x y z Ji r Ei r Δ i i x, y, z r i x, y, z J E J ( r E ( r eˆ ( r eˆ ( r J ( r E ( r cos ϕ Ji ( r Ei ( r (69 here T f ; the E i and J i are the magnitudes of E i and E J J i ; Δ ϕi ϕi ϕi, and the E J ϕ i and ϕ i are respectively the phases of E i and J i ; the e ˆE and e ˆJ are the direction vectors of E and J ; ˆ E i x, y, ziei, and E E E ; the J and J can be similarly expressed. Based on the (69 and that J E, the following relation (7 is derived. T act Re { } ( r, t dsdt T V T ( r, t dt ds T J ( r E ( r ds (7 In (69, the first inequality takes the mark of equality, if and only if Δ ϕ i ( r for any i x, y, z, i.e., the Ei ( rt, and Ji ( rt, have the same initial phase for any i x, y, z ; the second inequality takes the mark of equality, if and only if J E eˆ ( r eˆ ( r, i.e. ( r J ( r. Obviously, above two conditions are equivalent to that E ( rt, J ( rt, at any time t for a certain point r. In (7, the inequality takes the mark of equality, if and only if E ( rt, J ( rt, holds for any ( rt,, and this corresponds to the most efficient way that the E works on J, i.e., the most strongest iation state of J. In addition, Im{ } Im J ( r E ( r ds Im J ( r E ( r ds (7 Ji ( r Ei ( r sin ϕi( r ds V Δ i x, y, z Based on the (69-(7, it is clear that the most efficient way that the tterer iates needs that sin Δ ϕ i ( r for any r V and any i x, y, z, and then needs that Im{ } (i.e., resonance. owever, the resonance is not the sufficient condition to guarantee the most efficient iation state, because it cannot be guaranteed that the sin Δ ϕi ( r has the same sign for any r V and any i x, y, z. From above observations, the following conclusions are derived. The most complete descriptions for the tterer s ability to iate energies (i.e., the effectiveness of the interaction between E and tterer are the distribution of cosδ ϕi ( r and the distribution of ( Ji ( r Ei ( r on V, i.e., the distribution of Re{ ( J i ( r Ei ( r } on V, here i x, y, z. Of cause, the above mathematical descriptions can also be more physically depicted as follows. The effectiveness of the interaction between E and J depends on that: (. the distributions of the phase relations between the components of E and J ; (. the distributions of the magnitudes of E and J ; (. the distributions of the polarizations of E and J. The resonance condition is only a necessary condition to guarantee the strongest iation of electromagnetic system, but not a sufficient condition. D. The discussions and comparisons for the EC-SEFIE-CMT and EC-EM-CMT. In this subsection, the core priple of EC-SEFIE-CMT is summarized at first, and then some necessary discussions for EC-SEFIE-CMT are provided. The core priple of EC-SEFIE-CMT The EC-SEFIE-CMT was built based on the EFIE-based MoM in []-[]. The main destination of EC-SEFIE-CMT is to construct a transformation from any mathematically complete basis function set { } b to the CM set { a } which satisfies the power orthogonality given by the formulations (7 and (-(4 in paper []. The EC-SEFIE-CMT achieves the above destination by decomposing the impedance matrix Z in terms of its real part R and imaginary part X, and then solving the generalized characteristic equation X a λ R a. In fact, realizing the destination in by using the method in relies on the following necessary conditions. (. ( a a Z a ; (. The R and X must be symmetric matrices to guarantee that Re{ ( a } a R a and Im{ ( a } a X a ; (. The R must be positive definite [6]. 4 To guarantee the conditions listed in, the EC-SEFIE-CMT is realized as follows. (4. To guarantee the condition (., the Z must be constructed by using inner product, and the basis function set and the testing function set must be the same, and a coefficient should be contained; (4. To guarantee the condition (., the Z must be constructed by using symmetric product, and the basis function set and the testing function set must be the same; (4. To guarantee the condition (., the tterer is required to iate some electromagnetic energy. To simultaneously achieve above (4. and (4., the EC-SEFIE-CMT expands the ttering current in terms of real basis function set [], and then the EC-SEFIE-CMT can only construct the real characteristic currents, because in the vector space the characteristic vectors derived from X a λ R a are real. The discussions for EC-SEFIE-CMT Firstly, the EC-SEFIE-CMT has not ability to provide the complex characteristic currents, but it is more suitable for some electrically large structures, such as the travelling wave

13 R. Z. LIAN: EM-BASED CMT FOR EC antennas, to depict their inherent characteristics by using the complex characteristic currents [5]. Secondly, the EC-SEFIE-CMT doesn t provide any efficient method to research the non-iative modes. Although some scholars have found that the CMs corresponding to the infinite characteristic values are just the non-iative modes [], [6]-[7], the infinite characteristic values are not the necessary conditions to judge the non-iative modes as explained in the Sec. VI of this paper. The discussions for the OutCM set in EC-EM-CMT At first, it must be clearly pointed here that the physical essence of the OutCM set constructed in this paper is the same as the CM set derived from the titional EC-SEFIE-CMT. owever, the former is more advantageous than the latter in the following aspects. In this paper, the put power matrix is decomposed in terms of two ermitian matrices, and, as illustrated in the formulation (, and they respectively correspond to the iated power and red power as proven in the (. This decomposition can always be realized, and doesn t need to restrict the basis function set { } b to be real. Based on this, the characteristic currents are not restricted to be real, and then this paper provides a possible way for constructing the complex characteristic currents. When the is positive definite, the OutCM set is obtained by solving the generalized characteristic equation a λ a in (4, and this equation is the necessary condition to orthogonalize the modal put powers as (45. When the is positive semi-definite at frequency f, the frequency f is efficiently recognized by employing the modal put impedance Z given in (4, and then the OutCM set at f is obtained by using a limiting method given in (44. In addition, the EC-EM-CMT is not only suitable for the tterer placed in vacuum like EC-SEFIE-CMT, but also valid for the tterer placed in any time-harmonic electromagnetic environment. It is found in this paper that for a tterer, which is regarded as a whole object, its various power-based CM sets are independent of the electromagnetic environment which surrounds it. The relation between EC-SEFIE-CMT and EC-EM-CMT Based on the above observations, the CM set derived from EC-SEFIE-CMT can be viewed as a special case of the OutCM set in EC-EM-CMT framework, when the following conditions are simultaneously satisfied. (a Only real characteristic currents are desired; (b there doesn t exist any non-iative mode; (c the tterer is placed in vacuum. In fact, a more valuable effect to treat the EC-SEFIE-CMT as a special case of EC-EM-CMT is to liberate the EC-SEFIE-CMT from its titional framework, IE-MoM, and this liberation is very important for CMT, as discussed in the following Sec. VIII-E. E. The discussions for the variants of EC-SEFIE-CMT under the IE-MoM framework. After that the physical validity of the EC-SEFIE-CMT is verified by some applications in antenna community [8]-[], some IE-MoM-based variants of EC-SEFIE-CMT are raised gually. Obviously, only the impedance matrix derived from the Galerkin-based EFIE-MoM for EC system has the same physical essence as the put power quadratic matrix as discussed in Sec. VIII-D. It has been pointed in [4] and [] that the power characteristic of the CM set derived from Mat-VIE-CMT is physically unreasonable. In fact, the physical validities of all variants for the EC-SEFIE-CMT under IE-MoM framework should be reexamined carefully. F. The discussions for the modal quantities introduced in this paper and the titional characteristic quantity Modal Significance (MS. Obviously, the various modal quantities discussed in the Sec. VII and the modal component c in the expansion formulation (6 satisfy the relation (7. mod, act In fact, the MAO in (66 is equivalent to the titional MS as illustrated in the following (7. Titional MS j λ Im Re Re Re j( { } { } Re{ } { } jim{ } { } MAO mod, act (7 here the second equality is due to the relation (4, and the third equality originates from that Re{ } > for the iative OutCMs. It is easily found from (7-(7 that the physical essence of the titional MS is to quantitively depict the modal ability to allocate the al modal put power to its active part, instead of a quantitive depiction for the modal weight in whole modal expansion formulation The essential reason leading to the above problem of the titional MS is carefully analyzed as below. When the CM M is normalized by using the normalization way given in [], i.e., the modal iated power is normalized to be unit, the normalized CM is specifically denoted as the symbol M to be distinguished from the normalized version M sys, sys, SMS SMS c j j act mod mod, act mod mod, SMS SMS MACE GMS MAO MACE GMS MAO mod mod, act mod mod, MACE GMS MAO j MACE GMS MAO sys, act sys, GMS GMS (7